↑

Properties

Label 2-136-17.15-c1-0-1
Degree $2$
Conductor $136$
Sign $0.911 - 0.411i$
Analytic cond. $1.08596$
Root an. cond. $1.04209$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.152 + 0.368i)3-s + (−1.09 + 0.454i)5-s + (4.72 + 1.95i)7-s + (2.00 − 2.00i)9-s + (−1.79 + 4.33i)11-s − 3.10i·13-s + (−0.335 − 0.335i)15-s + (−3.95 − 1.16i)17-s + (−3.07 − 3.07i)19-s + 2.04i·21-s + (0.603 − 1.45i)23-s + (−2.53 + 2.53i)25-s + (2.15 + 0.892i)27-s + (2.98 − 1.23i)29-s + (−1.26 − 3.06i)31-s + ⋯
L(s)  = 1  + (0.0882 + 0.212i)3-s + (−0.490 + 0.203i)5-s + (1.78 + 0.739i)7-s + (0.669 − 0.669i)9-s + (−0.541 + 1.30i)11-s − 0.862i·13-s + (−0.0865 − 0.0865i)15-s + (−0.959 − 0.281i)17-s + (−0.705 − 0.705i)19-s + 0.445i·21-s + (0.125 − 0.303i)23-s + (−0.507 + 0.507i)25-s + (0.414 + 0.171i)27-s + (0.555 − 0.229i)29-s + (−0.228 − 0.550i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15324 + 0.248138i\)
\(L(\frac12)\) \(\approx\) \(1.15324 + 0.248138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (3.95 + 1.16i)T \)
good3 \( 1 + (-0.152 - 0.368i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (1.09 - 0.454i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-4.72 - 1.95i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.79 - 4.33i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.10iT - 13T^{2} \)
19 \( 1 + (3.07 + 3.07i)T + 19iT^{2} \)
23 \( 1 + (-0.603 + 1.45i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.98 + 1.23i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.26 + 3.06i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.51 + 6.06i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (6.91 + 2.86i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (3.39 - 3.39i)T - 43iT^{2} \)
47 \( 1 + 1.50iT - 47T^{2} \)
53 \( 1 + (-1.24 - 1.24i)T + 53iT^{2} \)
59 \( 1 + (-2.56 + 2.56i)T - 59iT^{2} \)
61 \( 1 + (-10.3 - 4.30i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + (-3.11 - 7.52i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-8.17 + 3.38i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.507 - 1.22i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-3.71 - 3.71i)T + 83iT^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 + (3.27 - 1.35i)T + (68.5 - 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.14768489981284098659449640171, −12.18456759718824168086937382845, −11.29718813599270803042768474442, −10.32322755142329852040612513325, −9.008029612378328047225289789010, −7.993589137500043829872636458568, −7.00652852854244009641331260107, −5.18689080603789302812899271429, −4.28560046778308338083821984217, −2.20070328300330453091470007615, 1.72720760532143686865850720690, 4.09568432302561939117877911025, 5.03882977475266095896492405531, 6.82983658627350995508513674428, 8.113236460746105604737566612246, 8.424538590876911515907253785425, 10.41986653874837203426600506114, 11.08087019105713493544632986770, 11.96883773857951834221416929274, 13.45197151603563369587280461566

Graph of the $Z$-function along the critical line